Question

read the proof.
given: \\( \overline { a e } \perp \overline { e c } ; \overline { b d } \perp \overline { d c } \\)
prove: \\( \triangle a e c \sim \triangle b d c \\)
what is the missing statement in step 4?
\\( \angle a c e \cong \angle b c d \\)
\\( \angle e a b \cong \angle d b c \\)
\\( \angle e a c \cong \angle e a c \\)
\\( \angle c b d \cong \angle d b c \\)

Explanation:

Step1: Analyze the triangles

We are given \(AE\perp EC\) and \(BD\perp DC\), so \(\angle AEC=\angle BDC = 90^{\circ}\). To use the AA (Angle - Angle) similarity theorem (\(\triangle AEC\sim\triangle BDC\)), we need another pair of congruent angles.

Step2: Check the angles

The angle \(\angle ACE\) is common to both \(\triangle AEC\) and \(\triangle BDC\). By the reflexive property of congruence, an angle is congruent to itself. So \(\angle ACE\cong\angle BCD\) (since they are the same angle).

Answer:

\(\angle ACE\cong\angle BCD\)